Analyzer



Oct. 10, 1950 4 G, D. MGCANN 2,525,496

DemodU/afar n hun B11/yer Pham? 5/7//er wlTNESSEs: 3j y INVENTOR 554 //berD McCann.

. fM/, @dem ATTORNEY G. D. MGCANN Oct. l0, 1950 ANALYZER 3 Sheets-Sheet 2 Filed Sept. 28, 1946 INVENTOR //e/ /WfCa/yn,

BY @IM/. 7'"

ATTORNEY WITNESSES Patented Oct. 10, 1.950

ANALYZER Gilbert D. McCann, Pittsburgh, Pa., assgnor to Westinghouse Electric Corporation, East Pittsn burgh, Pa., a corporation of Pennsylvania,

Application September 28, 1946, Serial No. 7 00,130

8 Claims.

This invention relates generally to a method and/or means for electrically ascertaining or determining the response of a physical system to a known set of forcing functions acting thereon, whether such quantities are of a static or transient character.

Fundamentally, the invention is directed to a method and/or means for determining, on the basis of electrical analogy, the response or reaction of a physical system when subjected to a known set of conditions.

More specifically stated, the invention is dlrected to a method and/or means of electrical analysis of a physical system which provides for the production of electrical quantities indicative of the forcing functions which excite the physical system to be analyzed, which introduces the electrical quantities thus produced into an electrical network or system representing the electrical counterpart of the physical system Vto be analyzed, according to the electrical analogy of the manner in which the said forcing functions eX- cite the said physical system, and which thereafter ascertains, or provides for the determination of, the electrical reactions in the electrical counterpart of the said physical system.

In this basic concept the present invention is related to a copending application of G. D. Mc- Cann and I-I. E. Criner, Serial No. 564,831, entitled Analyzer, filed on November 23, 1944, and assigned to the same assignee as this invention, now U.v S. Patent 2,420,891, granted May 20, 1947.

The said copending application covers generally the methods of analysis by means of electrical vanalogue andto this end illustrates and discusses the various forms of the invention in its vapplication to numerous problems. While' thebasic principles of the invention in the said copending ticularly adapted to the solution of problems defined in terms of non-linear functions.

Still another and more specific object of this invention is to provide a system of analysis of the character mentioned which provides for a circuit system or network which in its various parameters and their relation in the circuit is the electrical counterpart of the physical system to be analyzed, which provides for the producltion of electrical quantities representative of known conditions under which the physical `system must operate, to energize said circuit system, which utilizes said electrical quantities in the excitation of said circuit system, andV which thereafter provides `an indication or a record of the response of the system -to saidA quantities.

A specific'object of this invention is to provide a system of analysis of the character referred to which is applicable in the solution lof various types of functions series. Y Y

Another specific object of this invention is to provide `a system of `analysis wherein systems def-ined bysets of simultaneous equations Yinvolving non-linear unilateralv coefficients or terms vmay be analyzed.

The foregoing statements are merelyvillustra'- tiveof the various aims and objects ofgthisinvention. Other `objects and advantages -will become apparent upon a study of thefollowing y `speciiication when considered in conjunction with the accompanying drawings, in which: Fig. 1 is a.b1ock diagram illustrating in .general way an electrical multiplying device. y

Fig. 2 is a diagrammatic showing of a preferred type of multiplying device.

Fig. 3 is a schematic diagram, detailing the circuit connections of the multiplier of Fig. 2.

Fig. 4 is a diagrammatic showing of a-circuit network for developing electrical quantities representative of the mathematical definition of a particular type of self-.impedance or more; genexpressable in terms of a )power t 3 rally self-coefficients of a mathematical equation.

Fig. 5 is a diagrammatic showing of a circuit network applying the principles of Fig. 4 in the solution of a specific example.

Fig. 6 is a diagrammatic showing of a, circuit network for representing any type of self-impedance or self-coeflicient in an algebraic or differential equation.

Fig. 7 is a diagrammatic showing of a circuit network based upon the mathematical definition of a non-linear unilateral impedance for producing Voltage in one circuit from another circuit, or in the mathematical equation to be solved, the unilateral coeiiicient.

Fig. 8 demonstrates the application of such a system as illustrated in Fig. 'l in a practical problem.

to briefly consider the application of the electrical analogy from the purely mathematical viewpoint using the mechanical-electrical analogy as an example. For a wide range of physical systems, the proper analogous circuit can best be determined by the establishment of a consistent analogy between the variables and constants of electric circuit theory and those of the physical system to be analyzed. This method has been covered in detail in the above-mentioned copending application. As an illustration analogies for mechanical vibratory systems are given in Table I which follows at the end of this paragraph. Familiarity with the analogiesrfrequently permits construction of the analogous circuits without the necessity of completely formulating the mathematical equations and is a great aid in the general analysis of the system.

Analogous electrical systems Mechanical system Mass-inductance circuit Mass capacitance circuit Voltage (V) =Force or Torque Current (l) =Force or Torque orce or Torque t Force or Torque Mass or Inertia Inductance (L) Inductance (L) Capacitance (0)..

Capacitance (C).

Velocity Damping.. Resistance (R) Resistance (R) Conductancc Conductance Spring Constant Susceptance Susceptancc Inverselnductance Inverselnductane Force or Torque Voltage (V) Current (I) v Displacement Charge (f Idt) (I) (f Vdt) (V) velocity (1).. ZTI (viT Acceleration g-I 3% v g-ZY The analyzer of this invention is an electrical analogue type of computer in which the physical system set up to satisfy the mathematical equations deiining'the system to be analyzed, is Acompriselv of electrical circuits. The algebraic or differential equations defining the physical*Y system to b e analyzed are, in the case of linear systems,V represented by electric circuit parameters such as resistors, inductors and capacitors. AFor linear-,systems involving negative impedances or energy sources, such as required for servomechanisms, that is, any self-regulating system such as a speed controller or angle-position controller, electronic ampliiers are used. Also, as will hereinafter be shown, special electric networlrshave been developed'for basic non-linear functions.

, Additionally,various types-of circuits have been developed, some ci which Aare illustrated in the ducing any arbitrary sets of excitation functions In practice all the func-V the system of electrical analogy, it might .be well '7`5 rIphe choice of analogy depends upon a number of factors. For most transient problems a direct record of the measured .variable as a function of time can be moreeasilyobtained with a cathode rayoscilloscope which isessentially` 'a voltage measuring device. Therefore, when possible, it

is best to use the analogy for whichthe desired quantity is proportional to a voltage. Table I thus becomes aA guide to determine the most usefulanalogy. y H

`lin setting up an` analogous electrical circuit by the relations of TableLifthe impedances and electrical.excitation functions, such as the sudden application of voltage are madeequal to the mechanical quantities they represent in consistent systems; of units' all electrical solutions will be numerically equal-to the true mechanical solutions. elo'wever,l itv is desirable to change the electrical circuit constants by a fixed ratio so that it is 4not necessary to provide an excessively wide range of variations in circuit components inthe coriiputinglY device. Frequently the time base must also be changed Vfor the same reason or so thatthe solution can berecorded more readily. A consistent set of conversion formula is given in Tablelfollowing this paragraph'.l ln practice, it is also usually more convenient to set the electrical variable representing the known excitation function at an arbitraryV value and record the solutions as ratios of this by the equations ci TableII. 'w r Table II.-C'onversion ,formula for circuit con- Where a is an arbitrary constant, n s the ratio of the frequencies in the analogous circuit to those in the actual system. The Ls, Cs, and Rs are respectively linductance, capacitance, and resistance, while M, G and k, respectively represent mass, damping characterv istics, and spring constant.

(l) If the known excitation function is represented by a voltage Eo', actual voltages or quantities they represent are glven by the following equations:

Eo F Y eFiTen fn=E-0,e Y and currents or their analogies are vrepresented in Vthe following equations: f

aEo. uI'ov n=d1nl Un==nfllnl (2) When the known excitation function is represented bycur rents 1u', then:

The analogous electrical circuits can also readily be developed from the abstract mathematical equations. This may be well illustrated in the method of setting up electrical circuits to represent any set of one or more simultaneous equations. The formation of equations, involving the process of addition, is readily accomplished with electrical circuits. By either of Kirchos two laws sets of currents or voltages, representing the equations term by term, can be added and equated. The equation for voltages is more commonly used and will be touched up'on here. Each term of the equation can be set up by connecting circuit elements in the proper relationship in an analogy network. Other mathematical process may be required. These include multiplication, division, the formation of integral or differential relations between the dependent and independthe connection inv suitable circuit configuration, usually a simple loop circuit ,for a second order i equation, of resistors, inductors and capacitors of proper value. In addition to thesev circuit elements suitable voltage sources are required capable of producing any arbitrary function of the independent variable, which, in this analogy, is With a single loop circuit, as mentioned,

time. only a second order equation can be formed.

However,- a single equation of any order can usu- Y For certain types of equations negative coefii cients are involved such as negative resistances. These may be self or mutual coefficients. Additionally unilateral coefficients occur. l

The extension of hte electrical analogy to non-` linear types of equations involves two additional types of circuit elements. One of these is a, multiplying device capable of multiplying any two variable voltages together and giving their instantaneous product. The other device is one capable of producing any arbitrary function of a dependent variable. The latter ofthe two is the more diflicult functional propertyto obtain vfor an electrical analogue type of computer when inductors Yand capacitors are used'to form the differential relations. This conditionexists because'of the relatively high frequencies at which all of the functional elements must operate Y(6() to 2000 cycles per second). Operation at much lower frequencies would require very large inductors and capacitors. y

This difficulty can be overcome by the use of electronic integrators or differentiators. Through the use of properly designed, high gain D. C. amplifiers in the integrating or differentiating circuits the speed' of response can be slowed downy to a point where a matter of minutes may be involved in obtaining the function. This considerably simplifies the multiplication and representation of both the independent and dependent variables and further makes the selection of a multiplying device simpler since a wider range of types may be employed including the mechanical variety.

The elements for producing arbitrary functions are of three general types all having very low output'impe'dance so as not to disturb the analogous circuit, this being arequirement in the analogy networks based on the force-voltage analogy. Steady state sinusoidal forcing functions are conveniently produced by variable frequency signal generators feeding amplifiers with very low output impedance. Another type of transient excitation function may be produced by a magnetic v tape recorder. The magnetic tape is run at slow speed and magnetized in accordance with any predetermined function. It is then run at any arbitrarily selected higher speed through a suitable pickup feeding associated amplification equipment if needed, to generate a voltage proportional to the desired function.l In athird type 'the arbitrary transient function is represented by separate fourier seriesrcomponents generated by discharginga series of capacitors through a suitable circuit network land adding up all the components by introducing all the currents into a commonlow impedance circuit electrically associated with the analogy circuit. This latter methody is disclosed in the cop'ending application 'hereinbefore mentioned and is more conveniently employed when the excitation function ,is given in Aterms of components as their addition and vplotting is not required.

Various types of multiplying devices have been considered for producing the product quantities appearing in some types of functions. At the present time, there are vthree types which are known to be feasible for this application. One is a magnetic oscille-graph wattrneter element re'- i'lecting light on a photocell so arranged that it puts out a current proportional to the angular deflection of the wattmeter element. This angular deflection is proportional to the product of the currents inkeach of the two coils of the element. The second type of. multiplier is "a wellknown rectox circuit used for single side l.band carrier. The third type and the one which is presently preferred is illustratedin Figs. 2 and 3Y of the drawing. This subject matter is covered in a copending application of R. C. Cheek, Serial No. 32,281, filed June 1l, 1948, now Patent No. 2,519,= 223, entitled Multiplier and assigned to the same assignee as this invention. While the subjects matter of Figs. 2 and 3 per se, form no part of this invention, it is herein presented that a better un= derstanding of the present invention may be had.

As illustrated in elementary form in Fig. 1, the purpose of the multiplier inA the electrical analogue type of computer is to form the product of two electrical quantities. The illustration is made in connection with a pair of voltages E1 and E2 applied across separate sets of input terminals, the product appearing across the output terminals being ikEiEz, in which expression ik represents the ampliiication factor of the amplier. It is required that the multiplier be relatively free of errors due to undesirable orders of curvature in the characteristics of the circuit elements employed. Previously suggested circuits have been based on direct multiplication of the two input functions in Vacuum tube circuits with supposedly square law characteristic curves. Although some available vacuum tubes have characteristic curves approaching the desired parabolic shape, third and higher orders of curvature are present in all in a, sufficient magnitude to introduce an undesirable amount ofdistortion and inaccuracy in the resulting product. In the .embodiment of Figs. 2 and 3, instead of being multiplied directly, the two quantities are multiplied in turn by an auxiliary or carrier Wave of a frequency which is high in comparison with the highest frequency component of the eX- pected product. The use of the carrier wave in this manner permits spurious -outputs due to undesired orders of curvature in the tube characteristics to be largely eliminated by conventional tuned circuits, which are tuned to the order of frequency of the true product of the input functions by the carrier Wave'. Spurious responses appearas harmonics oi the carrier frequency and as harmonics and intermodulation products of the components of the input functions and are bypassed by the tuned circuits, since they are of an entirely different order of frequency. y,

After the carrier Wave is multiplied in turn by each of the input functions, the carrier Wave is eliminated by a process of demodulation in a linear demodulator to which the product of the carrier wave and the input functions, plus an additional carrier frequency'wave of the proper phase position, is supplied. The resulting output is the product of the tvvo input functions alone.

in Fig. 2, the oscillator produces the carrier frequency Wave, e@ sin wit. This Wave is fed into a balanced modulator, one of the input functions fi(t) being used as the modulating signal. The output of the first balanced modulator is mif1 t e1 sin wit which is the product of the original carrier wave and the iirst input function fi(t) and the quantity mi is a constant.

The output of the iirst balanced modulator is of the same order of frequency as the original carrier Wave, and is used as the carrier wave in the second balanced modulator, to which the second input function f2(t) is supplied as the modulating signal. The resulting output which again visi proportional to vthe product of the modulating function by the carrier Wave itself suppressed, is mimaft) f2(t)61 Sin wit'. l

It can/be seen that the output of the second balancedlmodulatorgis proportional tc the output Y that would be obtained from a single balanced 8 modulator if the product. of thetwoinput functions f1(t)f2(t) had been usedas a modulating signal. Furthermore, if a carrier wave of the proper phase position and of peak amplitude greater than the peak amplitude of the output of the second modulator is added to the output of the second modulatona resulting Wave Will be obtained which will be vof the form ulator introduces a phase shift ip to the said out-` put, which may then be Written m1m`2f1(?f)f2(t)r61 VSill (wit-Hp) AWhen the outputs oi the second balanced modu lator and the phase shifter arel combinedin the input to the linear demodulator the resultmay be expressed as As shown in the blockdiagram, therefore, the Wave resulting from the addition of the original carrier Wave, shifted in phase to compensate for any constant phase shift occurring in the modu lator circuits and the output of the second modulator, are applied to the linear demodulator or detector of a type conventionally used in the demodulation of amplitude-modulated waves. The output of the demodulator is thus f1(t)f2(t).

The schematic diagram of Fig. 3 is based upon the block-diagram of Fig. 1. The various cornponents or stages of the multiplier carry legends corresponding to those of Fig. 2 and it is believed that this schematic diagram will be understood in connection "with the discussion previously made. v The buffer stages are provided primarily to prevent the feedback :of quantities which might disturb 'the' operation of' the oscillator. These may be considerednpart of the oscillator. The phase shifter may be of any conventional type Which Will provide a range of`var`iation in the phase of the carrier signal from zero to degrecs.` For a more detailed discussion of the system of Figs. 2 and .3, reference. may be had 'to the above-identi'ed cozpending application of R. C. Cheek.

Now this vinvention isl concerned with systems for the production of electrical quantities representative of non-linear Vfunctions and the use of such systems in the electricalY analogue type of computer or analyzer hereinbefore generally discussed, the specific object being to provide a suitable method and means fork analyzing several veryimportant classes oignon-linear-problems which are so difcult to solverby conventional. mathematics that they can not be adequately treated.

By way of example, reference may be made to problems existingi'n non-linear circuits these include thek analysis of circuits With saturable reactors, transformer inrushcurrent `problems and the analysis of circuits With non-linear resistors to mention a few By the various electrical analogies to other physical systems such as mechanical vibrating systemsigheat flow, etc-.,- the non-linear elements of these other systems can be represented Ibysuitable electrical components and the range of application thereby extended.

An arbitrary function which might represent The voltage across such a resistor is E=R(z) z`=RoiiRoa1i2i RoazisiRoai: (2)

The system that will 'represent this self-impedance is shown in Fig. 4 and it is arranged to represent the power series of Equation 2 above term by term. The circuit network includes the three multipliers M1, M2 and M3 which may each be of the type specically referred to in Figs. 2 and 3, and for the purpose of this illustration let it be assumed that a current i is circulated in the network from an external circuit. Such a circuit may form a ipart of the complete network including such analogy circuit as may be necessary for the particular lproblemto be considered and to which the voltage E is to be applied. This will be considered hereinafter. The multipliers are connected in a circuit network so that the output of the iirst is applied to the second and so on and the network may be extended to include anyvnumber ofl terms of a power series, within practical limits of course. Resistors R3, R5 and R7, respectively connected across the outputs of multipliers M1, M2 and M3 may or may not be utilized in the circuit connections. They may be considered to represent the output impedance of the respective amplifiers or as being physically connected in the manner illustrated. Their use will depend upon the circuit constants needed for Iproper performance or to form the desired constants of the individual terms of the power series.

Considering first the case in which the resistance of the various resistors indicated is appreciable and may not be neglected, the rst Voltage is produced by the current i flowing in the circuit loop including resistors R1, R2, R3, R4', Rs, Re and Rv. Thus the first voltage E1 maybe written The second voltage appears across the output of multiplier M1 which multiplies the voltage drop across resistor R2 caused by the current i and which is applied to each of its two input terminals as indicated. The product voltage is, therefore ikiRzZz'Z and the second voltage considering the drop across Rs caused by current i thus becomes This voltage is impressed across the upper pair oi input terminals of multiplier M2. The current z owing through the resistor R4 produces a voltage drop expressed by Rizl which is applied to the lower pair of input terminals on the multiplier Mz and the third voltage E3 becomes in a similar manner, the productof voltages E3 and Rsi which are respectively applied to the input terminals of multiplier M3 and which forms the fourth voltage of the series, is expressed'as a voltage Ei which is E4:iC1C2C3R22R4Rsi4ikzksRsR-iRsii kaRsRsz'z-i-R'n' The voltage E may now be written upon collecting the terms as in which the constant (Ri-l-Rz-kRa-I-Ri-l-Rs-I-Rs-I-Rv) corresponds to the constant Ro of Equation 2k above; the constant (MR22 i- C2R3R4 ik3R5Rs) corresponds to the constant Roar in (2), etc. It should be noted that R2, R4 and Rs may be only one resistor as all voltages may be taken from the single resistor R2 under certain conditions* The constants may be lumped inwhichvcase:

E=K1iiK2i2iKai3iK4i4i series involving R3, R5 andRv will be dropped and v the element voltages representing the power series term by term may be written as follows:

AIf Rz,`R4 and R6 are also suiciently smallltov be neglected as may be the caseA if the multiplier E represented by E=R1iik1a22i2iklkzRzZRiii Y kikzksRzzRiReili which may also be written E=K17iK22iK3i3iK44i To illustrate the application of the network of Fig. 4, consider the differential equation where, to continue the discussion in terms of the illustration of Fig. 4,

will be assumed to represent the non-linear characteristic of the resistor and lcz) being defined amgfR@chiariRoaleRoaziiRoaei The circuit for solving Equation 4 is shown in Fig.v5 in which the network of Fig. 4 is represented as a block designated MiMz'Ms. In this analogy, the first term of Equation 4,

is represented in the inductor L, the inductance of which is proportional to k1 as indicated. This is arranged in conjunction with the network MiMzMs as shown above. This current circulates through the analogy circuit ilo-wing through the multiplier network as indicated in Fig. 4 and produces the Various' functions across the resistors which are multiplied in the multiplying components of the network, together with other quantities embodied in the power series dening the Voltage of Equation 3. The voltage e used to excite the system may be produced by any of the methods generallydiscussed earlier in this disclosure and may represent a single or a multiplicity of components or other type `of variable depending,r upon the character of the function to be represented.

A circuit for representing any other type of self-impedance is shown in Fig. 6. Here the output impedance of the multipliers is presumed to be low. and is therefore neglected. It will be noted that the general circuit arrangement corresponds to that illustrated in Fig. 4; 'Ihus again the circuit `network provides for the term by term representation of a power series in terms of voltages. In some cases, the impedances Z2, Z3. and Z4 may have to be negligibly small for the rst term of the equation. This can be taken care of to a large extent'by a proper control or selection of the amplication of the multipliers. If conditions permit all the impedances may be the same in which case Z1, Z3 and Z4 may be eliminated and voltages for the associated multiplier input terminals may be obtained from Z2. Under other conditions. the various impedances may be of diierent type, in which case some may be:

e@ or Ri others may be dzq dI or Lgt 12 Y across the circuit terminals of the network of Fig. 6 will be expressed as If Z2, Z3 and Z4 are sufficiently small to be neglected then E iZiiikiZzZiz ilflikzrZzZZsi3 iIt1 J27 33Z22Z3Z4fL In each case E will be proportional to some nonlinear property of a particular problem to be investigated as dened by the equation representing that problem.

In the mathematical analysis of certain prob- Y lems terms frequently appear in the equations involving non-linear unilateral coeicients. This condition exists in the following typical set of simultaneous Equations (5) and (6) fr0) :klagt-163951 (5') in which K2 (xi) is assumed to be a non-linear function and is termed a non-linear unilateral coefcient since it appears in Equation (6) only, the term Y a di ducing the non-linear unilateral impedance, the

complete electrical solution of the set of simultaneous Equations (5) and (6) being set Aforth in Fig. 8.

Let it be assumed that the coeflicient K2(1) may be expressed as' is to be represented as a voltage to be produced. Then by the analogy the current to be circulated in circuit No. I of Fig. 7 will be proportional to iKiZlii where iK1 represents the amplification factor. This it will be noted is a Voltage say E1 representing the rst term of the power series deiining the total voltage E and is applied to circuit No. 2. The quantity Z1i1 is applied to both input terminals of multiplier Mi and, hence, the volt- Y13 age E2=iK2Z12ii2 and represents the second term of the power series. E2 is also vapplied to circuit No. 2. In producing the remaining term of the power series, the quantity'i '-I L2Z12ii2 is applied to one set of input terminals of multiplier M2 while the quantity Z1z'1 is yapplied across'theremaining set of input terminals. The-output of multiplier M2 thus becomes l -K3Z13z'13 which equals E3. The sum of the voltages around the loop of circuit No. 2 is thus equal to E in view of the term by term representation of the power series in the various voltages produced.

It is to be understood that the circuit network of Fig. 7 may be subjected to change to t particular problems. It may be extended o1' decreased to provide more or fewer'terms of a power series. The inputs to the various multipliers may be rearranged -toi'provide substantially any desired type of input function and for this purpose need not in all cases be directly connected into the circuit No. I as indicated. If the three impedances designated Z1 are of similar character and impedance requirements of circuit No. I Yand the input requirements ofY the multipliers and amplifier permit, the impedancesmay be lumped and suitable voltages takenv therefrom-to be applied to the multiplier V and amplier, the main consideration being the basic circuit congura'- tion to produce the desired arbitrary function,

which in this illustration is represented in voltages, in circuit No. 2. v

The complete solution of the set of simultaneous Equations (5) and (6) appears in Fig. 8

wherein circuits No. I and No. 2 are completed v The loop circuit No. 2 is completed by the addition of an inductor L2, a capacitor C2 and a voltage producing source e2,A in which:

The excitation functions represented in the voltages e1 and e2 maybe of any arbitrarily selected value in compliance with the' constant of'proportionality selected for the analogy and represent the known forces lto which the system to be analyzed is subjected. Y H

In Equation (5) the rst term of the righthand member is represented by the-voltage dropv across the inductor L1 and the remaining ter-m is represented by the voltage-on the capacitor C1.

In Equation (6) the -firstterm of the right-hand member is represented by thevvoltage dropacross the inductor L2. The second term is represented by the voltage produced in circuit. No. 2 by the amplifier and multiplier network and the remaining term is represented by the voltage across capacitor C2.

Referring to circuit No. I of Fig. 8 it will be noted that the current i1 whichr is proportional to i@ di flows through'the impedances Zi in that circuitv loop. Since lthis current flowing through the impedances will produce voltages in the No. I circuit loop which lwill appear as an additional term, it is essential, for this particular problem, that the impedances Z1 be suliciently small that the voltages produced thereacross may be neglected when the terms of Equation (1) are being calculated. Thus the input impedances of the amplifier and multipliers must be very low and it will be necessary in most cases to provide a fair amount of gain thereacross to produce output voltage. It has been possible to dc this thus far by conventional means.

Likewise to avoid disturbing the operation of circuit No, 2 the output of the amplier and multiplier network must be of low impedance. If,A for some reason, it is found that the amplier and multiplier network cannot be made of suiiiciently low impedance to meet the circuit requirements, the output impedance may be bucked out by the use Aof a negative impedance such as an ampliiier.

In the solution of various physical problems by the method of electrical analogy considerable variation in the basic circuits is usually required. However, this has been shown in this disclosure by the various illustrations which demonstrate the underlying principles involved by means of a few specific examples. The setting up of a problem for solution by the analogy method requires a consideration of the mathematics and circuit theory involved as well as knowledge of the physical systems of each iield of engineering to which the method is to be applied. For this reason', it is intended that the foregoing disclosure be considered only illustrative and not interpreted in a limiting sense. The only limitations are to be determined from the scope of'the appended claims.

I claim as my invention:

1. Apparatus for electrically representing a plurality of terms of a power series comprising, in combination, a plurality of electrical multiplying devices each having two pairs of input terminals to which the functions to be multiplied are applied and a pair of output terminals, circuit means connecting the output terminals of the first multiplier to one pair of input terminals of the second multiplier and connecting the output terminals of the second multiplier to one pair of input terminals of the third multiplier, and so on, a resistor network, means for exciting said resistor network, circuit means connecting said resistor network with said multiplying devices to apply at least a portion of the voltage drop therein to both pairs of input terminals of said first multiplying device and to the remaining pairs of input terminals of the remaining multiplying, devices, said resistor network producing a Voltage representative of the first term of said power series, said lfirst multiplying device producing a voltage representative of the second 'A term vof the power series and so on.

2; Apparatus for electrically representing la plurality of terms ofa power series comprising, in combination, a plurality of electrical multiplying devices each having two pairs of input l terminals to which the functions to be multiand so on, a resistor network,means for exciting said resistor network, circuit means connecting said resistor network with said multiplying devices to apply at least a portion oi the voltage drop therein to both pairs of input terminals of said first multiplying device and to the remaining pairs of input terminals of the remaining multiplying devices, said resistor network producing a voltage representative of the rst term of said power series, said rst multiplying device producing a voltage representation of the second term of the power series, and so on to the last multiplying device employed, and circuit means forming a part of the circuit system for said multiplying devices and said resistor network for combining the voltages of said resistor network and said multiplying devices according to the mathematical signs connecting the terms of said power series.

3. Apparatus for electrically representing a plurality of terms of a power series which comprises, an electrical network, a plurality of impedance elements forming a part of the network, means for exciting the network, a plurality of multiplying devices connected in the network, the rst to have applied thereto a pair of voltages appearing across at least a portion of the said impedance elements, the second multiplying device having applied thereto the output of the first multiplying device-and a voltage developed in said impedance network and so on to the last multiplying device, the voltage of said impedance network producing the rst term of said power series, the output voltage of the rst multiplying device producing the second term of the power series and so on to the last multiplying device.

Ll. YApparatus for electrically representing a plurality of terms of a power series which cornprises, an electrical network, a plurality of impedance elements forming a part of the network, means for exciting the network, a plurality of multiplying devices connected in the network, the first to have applied thereto a pair of voltages appearing across at least a portion of the said impedance elements, the second multiplying device having applied thereto the output of the first multiplying device and a voltage developed in said impedance network and so on to the last multiplying device, the voltage of said impedance network producing the rst term of said power series, the output voltage of the first multiplying device producing the second term of the power series and so on to the last multiplying device, and circuit 'means forming a part of the said electrical'network for combining the said voltages to form a single voltage according to the mathematical signs connecting the terms'of the said power series.

5. Apparatus for electrically representing a set of simultaneous equations, one of which involves a non-linear unilateral term which term is expressable in at least two terms of a power series comprising, in combination, a rst circuit including electrical elements representing term by term the equation without'the non-linear unilateral term, a second circuit including electrical elements representing term by'term the equation havingthe non-linear unilateral term excepting the non-linear unilateral term, excitation means for supplying a current to said first circuit, excitation means for supplying a current to said second circuit, impedance means electrically connected in said nrst circuit to be energized in dependence of the current owing therein, voltage amplication means electrically connected to age of said voltage multiplying means being theV product of the voltages at the input circuits thereof and representing the third term of 'said power series.

6. Apparatus for electrically representing a set of simultaneous equations, one of which involves a non-linear unilateral term which term is expressable in at least three terms of a'power series comprising, in combination, a rst circuit' including electrical elements representing term by term the equation without the non-linear unilateral term, a second circuit including electrical elements representing term by term the equation having the non-linear unilateral term excepting the non-linear unilateral term, excitation means for supplying a current to said rst circuit, excitation means for supplying a current to said second circuit, impedance means electrically connected in said rst circuit to be en` ergized in dependence of the current therein, voltage amplification means electrically connected to said impedance means to be energized in dependence of the voltage thereof and having the output thereof electrically connected in said second circuit, the output voltage of said voltage amplication means representing the first term of said power series, a first electrical voltage multiplying device having two input circuits and an output circuit, circuit means electrically connecting both input circuits to said impedance means, circuit means connecting said output circuit in said second circuit, theA output voltage of said rst voltage multiplying device being the product of the voltages at theinput terminals thereof and representing the second term of said power series, a second voltage multiplying device having two input circuits and an output circuit, circuit means electrically connecting one of said input circuits of said second voltage multiplying device to said impedance means, circuit means electrically connecting the remaining input circuit of said second voltage multiplying device to the-output terminals of said iirst voltage multiplying device, circuit means electrically connecting the output circuit of said second voltage multiplying device in said second circuit, the output voltage of said second voltage multiplying device being the product of the voltage at the input terminals thereof and representing the third term of said power series.

7; Apparatus for electrically representing a set of simultaneous equations, one of which involves a non-linear unilateral term which term is expressable in at least two terms of a power series comprising, in combination, a first circuit Lincluding electrical elements representing term by term the equation without the non-linear unilateral term, a second circuit including electrical elements representing term by term the equation having the non-linear unilateral term excepting the non-linear unilateral term, circuit means for producing ar current in said first circuit, circuit means for producing a current in said second circuit, a first electrical voltage producing device having its input electrically connected to said impedance means and its output electrically connected in said second circuit for producing a voltage in said Second circuit corresponding to the rst term of said power series, and a second electrical Voltage producing device having its input electrically connected to 'said impedance means and its output electrically connected in said second circuit for producing a Voltage in said second circuit corresponding to the second term of said power series.

8. Apparatus for producing an electrical quantity representative ofa non-linear unilateral term of a pair of simultaneous equations, said term being expressable in at least two terms of a power series, said apparatus comprising, a rst circuit electrically representing term by term the terms of the equation without the non-linear unilateral term, means vfor producing a current in said rst circuit, first circuit means responsive to the current in said rst circuit for producing a Voltage corresponding to the first term of said power series, second circuit means responsive to the current in said rst circuit for producing a voltage corresponding to the Second term of said power series, and an electrical circuit for combining said voltages for producing a voltage representative of said non-linear unilateral term.

GILBERT D. MCCANN.

REFERENCES CITED The following references are of record inA the y le of this patent: y

UNITED STATES PATENTS OTHER REFERENCES An Electro-Mechanical Methodfor Solving Equations by A. I-I. Schooley,-RCA Review, Vol. III, No. 1, July v1938, published by RCA Institute Technical Press, New York, N. Y., pages 86-96. 

